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Neural Computing and Applications

, Volume 31, Issue 10, pp 6039–6054 | Cite as

Projective synchronization for fractional-order memristor-based neural networks with time delays

  • Yajuan Gu
  • Yongguang YuEmail author
  • Hu Wang
Original Article

Abstract

In this paper, the global projective synchronization for fractional-order memristor-based neural networks with multiple time delays is investigated via combining open loop control with the time-delayed feedback control. A comparison theorem for a class of fractional-order systems with multiple time delays is proposed. Based on the given comparison theorem and Lyapunov method, the synchronization conditions are derived under the framework of Filippov solution and differential inclusion theory. Several feedback control strategies are given to ensure the realization of complete synchronization, anti-synchronization and the stabilization for the fractional-order memristor-based neural networks with time delays. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.

Keywords

Memristor-based neural networks Synchronization Fractional-order Time delays 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant (No. 11371049 and No. 61772063) and the Fundamental Research Funds for the Central Universities under Grant No. 2017YJS194.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Chua L (1971) Memristor-the missing circuit element. IEEE Trans Circuit Theory 18(5):507–519Google Scholar
  2. 2.
    Strukov D, Snider G, Stewart D, Williams R (2008) The missing memristor found. Nature 453(7191):80–83Google Scholar
  3. 3.
    Anthes G (2011) Memristors: pass or fail ? Commun ACM 54:22–24CrossRefGoogle Scholar
  4. 4.
    Wu A, Zeng Z (2012) Exponential stabilization of memristive neural networks with time delays. IEEE Trans Neural Netw Learn Syst 23(12):1919–1929CrossRefGoogle Scholar
  5. 5.
    Zeng Z, Wang J (2008) Design and analysis of high-capacity associative memories based on a class of discrete-time recurrent neural networks. IEEE Trans Syst Man Cybern Part B: Cybern 38(6):1525–1536CrossRefGoogle Scholar
  6. 6.
    Zeng Z, Wang J (2009) Associative memories based on continuous-time cellular neural networks designed using space-invariant cloning templates. Neural Netw 22:651–657zbMATHCrossRefGoogle Scholar
  7. 7.
    Zeng Z, Wang J, Liao X (2004) Stability analysis of delayed cellular neural networks described using cloning templates. IEEE Trans Circuits Syst I: Regul Pap 51(11):2313–2324MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Zhang G, Shen Y, Quan Y, Sun J (2013) Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays. Inf Sci 232:386–396MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Yang X, Cao J, Yu W (2014) Exponential synchronization of memristive Cohen–Grossberg neural networks with mixed delays. Cogn Neurodyn 8:239–249CrossRefGoogle Scholar
  10. 10.
    Li N, Cao J (2015) New synchronization criteria for memristor-based networks: adaptive control and feedback control schemes. Neural Netw 61:1–9zbMATHCrossRefGoogle Scholar
  11. 11.
    Pecora L, Carroll T (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8):821–824MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bao H, Park JH, Cao J (2016) Exponential synchronization of coupled stochastic memristor-based neural networks with time-varying probabilistic delay coupling and impulsive delay. IEEE Trans Neural Netw Learn Syst 27(1):190–201MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mainieri R, Rehacek J (1999) Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 82(15):3024–3045CrossRefGoogle Scholar
  14. 14.
    Chee C, Xu D (2006) Chaos-based M-ary digital communication technique using controller projective synchronization. IEE Proc Circuits Dev Syst 153(4):357–360CrossRefGoogle Scholar
  15. 15.
    Wang S, Yu Y, Diao M (2010) Hybrid projective synchronization of chaotic fractional order systems with different dimensions. Phys A 389(21):4981–4988CrossRefGoogle Scholar
  16. 16.
    Wang S, Yu Y, Wen G (2014) Hybrid projective synchronization of time-delayed fractional order chaotic systems. Nonlinear Anal: Hybrid Syst 11:129–138MathSciNetzbMATHGoogle Scholar
  17. 17.
    Peng G, Jiang Y, Chen F (2008) Generalized projective synchronization of fractional order chaotic systems. Phys A 387(14):3738–3746CrossRefGoogle Scholar
  18. 18.
    Zhou P, Zhu W (2011) Function projective synchronization for fractional-order chaotic systems. Nonlinear Anal-Real 12(2):811–816MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Chen L, Chai Y, Wu R (2011) Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems. Phys Lett A 375(21):2099–2110zbMATHCrossRefGoogle Scholar
  20. 20.
    Park JH (2008) Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters. J Comput Appl Math 213(1):288–293zbMATHCrossRefGoogle Scholar
  21. 21.
    Park JH (2007) Adaptive modified projective synchronization of a unified chaotic system with an uncertain parameter. Chaos, Solitons, Fractals 34(5):1552–1559zbMATHCrossRefGoogle Scholar
  22. 22.
    Lundstrom B, Higgs M, Spain W, Fairhall A (2008) Fractional differentiation by neocortical pyramidal neurons. Nat Neurosci 11(11):1335–1342CrossRefGoogle Scholar
  23. 23.
    Kaslik E, Sivasundaram S (2011) Dynamics of fractional-order neural networks. In: Proceedings of the international conference on neural networks. California, USA, IEEE, p 611–618Google Scholar
  24. 24.
    Kaslik E, Sivasundaram S (2012) Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw 32:245–256zbMATHCrossRefGoogle Scholar
  25. 25.
    Yu J, Hu C, Jiang H (2012) \(\alpha\)-stability and \(\alpha\)-synchronization for fractional-order neural networks. Neural Netw 35:82–87CrossRefzbMATHGoogle Scholar
  26. 26.
    Song C, Cao J (2014) Dynamics in fractional-order neural networks. Neurocomputing 142:494–498CrossRefGoogle Scholar
  27. 27.
    Zhang S, Yu Y, Wang H (2015) Mittag-Leffler stability of fractional-order Hopfield neural networks. Nonlinear Anal: Hybrid Syst 16:104–121MathSciNetzbMATHGoogle Scholar
  28. 28.
    Chen L, Chai Y, Wu R, Ma T, Zhai H (2013) Dynamic analysis of a class of fractional-order neural networks with delay. Neurocomputing 111(2):190–194CrossRefGoogle Scholar
  29. 29.
    Wang H, Yu Y, Wen G, Zhang S (2015) Stability analysis of fractional-order neural networks with time delay. Neural Process Lett 42(2):479–500CrossRefGoogle Scholar
  30. 30.
    Wang H, Yu Y, Wen G (2014) Stability analysis of fractional-order Hopfield neural networks with time delays. Neural Netw 55:98–109zbMATHCrossRefGoogle Scholar
  31. 31.
    Wang H, Yu Y, Wen G, Zhang S (2015) Global stability analysis of fractional-order Hopfield neural networks with time delay. Neurocomputing 154:15–23CrossRefGoogle Scholar
  32. 32.
    Bao H, Park JH, Cao J (2015) Adaptive synchronization of fractional-order memristor-based neural networks with time delay. Nonlinear Dyn 82(3):1343–1354MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Yu J, Hu C, Jiang H (2014) Projective synchronization for fractional neural networks. Neural Netw 49:87–95zbMATHCrossRefGoogle Scholar
  34. 34.
    Chen J, Zeng Z, Jiang P (2014) Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw 51:1–8zbMATHCrossRefGoogle Scholar
  35. 35.
    Bao H, Cao J (2015) Projective synchronization of fractional-order memristor-based neural networks. Neural Netw 63:1–9zbMATHCrossRefGoogle Scholar
  36. 36.
    Podlubny I (1999) Fractional differential equations. Academic Press, San DiegozbMATHGoogle Scholar
  37. 37.
    Kilbas A, Srivastava H, Trujillo J (2006) Theory and applications of fractional differential equations. Elsevier, New YorkzbMATHGoogle Scholar
  38. 38.
    Lakshmikantham V, Leela S, Devi J (2009) Theory of fractional dynamic systems. Cambridge Scientific Publishers, CambridgezbMATHGoogle Scholar
  39. 39.
    Li C, Deng W (2007) Remarks on fractional derivatives. App Math Comput 187:777–784MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Bhalekar S, Gejji V (2011) A predictor–corrector scheme for solving nonlinear delay differential equations of fractional order. Fract Calc Appl Anal 1(5):1–9Google Scholar
  41. 41.
    Aubin J, Frankowska H (1990) Set-valued analysis. Birkhäauser, BostonzbMATHGoogle Scholar
  42. 42.
    Filippov A (1988) Differential equations with discontinuous right-hand side. Kluwer Academic Publishers, DordrechtzbMATHCrossRefGoogle Scholar
  43. 43.
    Aguila-Camacho N, Duarte-Mermoud M, Gallegos J (2014) Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 19:2951–2957MathSciNetCrossRefGoogle Scholar
  44. 44.
    Gu Y, Yu Y, Wang H (2016) Synchronization for fractional-order time-delayed memristor-based neural networks with parameter uncertainty. J Franklin Inst 353:3657–3684MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Zhao J, Wang J, Park JH, Shen H (2015) Memory feedback controller design for stochastic Markov jump distributed delay systems with input saturation and partially known transition rates. Nonlinear Anal Hybrid Syst 15:52–62MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2018

Authors and Affiliations

  1. 1.Department of MathematicsBeijing Jiaotong UniversityBeijingPeople’s Republic of China
  2. 2.School of Statistics and MathematicsCentral University of Finance and EconomicsBeijingPeople’s Republic of China

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